A COMPARISON BETWEEN IMPLICIT AND EXPLICIT SPACECRAFT GYRO CALIBRATION
ITZHACK Y. BAR-ITZHACK
Faculty of Aerospace Engineering
Technion-Israel Institute of Technology
Haifa, 32000
ISRAEL
RICHARD R. HARMAN
Flight Dynamics Analysis Branch, Code 595
NASA-GoddardSpaceFlightCenter
Greenbelt, MD20771
USA
Abstract: - This paper presents a comparison between two approaches to sensor calibration. According to one approach, called explicit, an estimator compares the sensor readings to reference readings, and uses the difference between the two to estimate the calibration parameters. According to the other approach, called implicit, the sensor error is integrated to form a different entity, which is then compared with a reference quantity of this entity, and the calibration parameters are inferred from the difference. In particular this paper presents the comparison between these approaches when applied to in-flight spacecraft gyro calibration. Reference spacecraft rate is needed for gyro calibration when using the explicit approach; however, such reference rates are not readily available for in-flight calibration. Therefore the calibration-parameters estimator is expanded to include the estimation of that reference rate, which is based on attitude measurements in the form of attitude-quaternion. A comparison between the two approaches is made using simulated data. It is concluded that the two approaches yield comparable results but the implicit software implementation is less complex than the explicit implementation.
Key-Words: - Calibration, Gyroscopes, Spacecraft, Kalman-Filter
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1 Introduction
In-flight sensor calibration is a crucial element in spacecraft preparation for a successful mission. During the first stage of the calibration process the sensor errors sources, which are called calibration parameters, are estimated. During the second stage, errors caused by these calibration parameters are removed from the sensor readings.
There are two approaches to sensor calibration; namely, explicit and implicit. In the explicit approach an estimator compares the sensor readings to reference readings, and it uses
the difference between the two to estimate the calibration parameters. According to the implicit approach the sensor error is integrated to form a different entity, which is then compared with a reference quantity of this entity, and the calibration parameters are inferred from the difference. If, for example, the sensors are gyros, then using the explicit approach estimator compares the gyro readings to a reference rate. If the implicit approach is applied, then the gyro readings are properly integrated to generate attitude and the latter is compared with the reference attitude. In both approaches the difference is used by the estimator, which estimates the calibration parameters. Obviously, reference rate is required in the explicit approach to gyro calibration, and reference attitude is needed when the implicit approach is used.
An example for the use of the explicit approach can be found in [1] and an improvement of it is presented in [2] where the calibration of the AQUA satellite gyroscopes was treated. Since the calibration was done in-flight, a reference angular rate was not readily available. Therefore measured attitude was used to estimate the reference angular rate vector. Subsequently an estimator used the difference between the estimated reference rate and the gyro-measured rate to estimate the calibration parameters. Actually, the two estimators were combined into one Kalman filter. Because the rate estimation was based on Euler’s equation that describes the spacecraft (SC) angular dynamics, the rate estimation was nonlinear which gave rise to the use of the Pseudo-Linear Kaman Filter (PSELIKA) [3]. The implicit approach is normally used for calibrating the inertial measuring unit in inertial navigation systems [4].
The purpose of the present work is the comparison between the explicit and the implicit approaches using simulated gyro data.
2 The Gyro Error Model
The gyro errors that are considered in this work are: misalignment, scale factor error, and bias (constant drift rate). The gyro error model is a linear model, which associates small error sources to the gyro outputs. Due to the linearity of the model we can compute the contribution of each error source independently and then sum up all the contributions into one linear model. We start the description of the error model, by deriving the expression for the gyro misalignments.
2.1 Misalignment Model
It can be shown [1] that the rate error due to misalignment of a body mounted gyro triad is:
(1)
where is the projection of the i-gyro sensitive-axis on the body-axis. Since the sensitive axis is described by a unit vector and due to the proximity of the gyro sensitive axis to the respective body-axis, is a misalignment angle, assumed to be small. The elements are the angular velocity components. Let,
(2)
and
(3)
where denotes the transpose. Then Eq. (1) can be rewritten as,
(4)
2.2 Scale Factor Error Model
As mentioned, other error sources that causes the difference between the correct value of the actual rates and their measurements are the scale factor errors. The error model for the scale factor errors is simply
(5)
where is the scale factor error of gyro . Eq. (5) can be written as follows
(6)
Define
(7.a)
and
(7.b)
then Eq. (6) can be written as
(7.c)
2.3 Bias Model
The bias error model is quite simple and is given by
(8)
where is the three dimensional identity matrix, and
(9)
and x, y, z are the corresponding gyro axes.
2.4 The Augmented Gyro Error Model
The total gyro error is the sum of all the error discussed before; namely, misalignment, scale factor and bias errors; that is
(10.a)
Using Eqs. (4), (7.c), and (8)
(10.b)
The last equation can be written in the following form
(10.c)
Define as follows
(10.d)
also let
(10.e)
then Eq. (10.c) can be written as
(10.f)
3 The Explicit Gyro Calibration Algorithm
The information that we have in-flight is attitude, and rates measured by the gyros but, unlike ground calibration, the reference rates needed for explicit calibration are not readily available. Therefore, as mentioned earlier, in this case we have to estimate the angular rate vector while estimating the calibration parameters. We use then the attitude information to estimate the angular rate. The attitude information can be supplied in various ways; namely, it can be in the form of raw vector measurements or it can be given in an already processed form as attitude quaternion, for example. The estimation of the angular rate vector hinges on Euler’s equation that describes the SC angular dynamics as specified below:
(11)
where I is the SC inertia tensor, is the cross product matrix of the vector , is the angular momentum of the momentum wheels, and is the external torque operating on the SC.
As mentioned before, the reference rate is estimated from the measured attitude. This necessitates the inclusion of the attitude dynamics in the estimator dynamics equation. Suppose that the attitude is measured by autonomous star trackers (AST) [5] that yield the measured quaternion. Its dynamics equation is expressed by [6]
(12.a)
where
(12.b)
Equation (12.a) can be also written as
(12.c)
where
(12.d)
Because (Eq 10.e) is a constant vector, it obeys the following differential equation
(13)
We can combine Eqs. (11), (12.c) and (13) into one dynamics equation, add white noise to the angular dynamics equation to account for model uncertainty, add a small quantity of white noise to the calibration parameter dynamics to better model them in case they are not really constant but rather vary slowly, and add white noise to the quaternion dynamics equation to account for modeling inaccuracies. As a result we obtain the following dynamics equation
… (14)
where is the noise vector added to the angular dynamics, is the noise vector added to the calibration parameters, and is the white noise added to the quaternion dynamics. Because the dynamics matrix is a function of as well as , the dynamics equation presented in Eq. (14) is nonlinear, subsequently a nonlinear estimator is needed for estimating the state vector. The most appropriate estimator for the nonlinearity structure of this dynamics model is the PSELIKA algorithm [2].
As explained before, we have two measurements to consider; namely, the gyro rate-measurements, , and the quaternion measurement, . The former consists of the true rate, the error due to the calibration parameters, , and some high frequency noise which is modeled as a zero-mean white noise, . Consequently we have
(15.a)
Using Eq. (10.f), the last equation can be written in the following matrix form
(15.b)
The quaternion measurement is modeled by a combination of the true quaternion and a zero-mean white measurement noise vector, .
(15.c)
As explained in the introduction, we combine the estimators that estimate the calibration parameters and that which estimates the rate vector into one estimator; therefore we have to properly combine the two measurement equations. The combination of Eqs. (15.b) and (15.c) yields
(15.d)
Equations (14) and (15.d) constitute, respectively, the dynamics and the measurement equations for the augmented estimator used in the explicit approach to gyro calibration.
4 The Implicit Gyro Calibration Algorithm
As in the previous case, our goal now is to estimate , and for that we need to know how influences the attitude estimation. The true quaternion obeys the differential equation (12.a) where is a function of the true angular rate vector, , which we do not know. We know though, and therefore we rather use, the measured rate vector, . According to Eq. (15.a), therefore
(16.a)
and similarly
(16.b)
where
(16.c)
and where and are functions of and respectively, and are in the format of Eq. (16.c). Using Eq. (16.b), Eq. (12.a) can be written as:
(17.a)
The latter can be written as
(17.b)
where Q is as in Eq. (12.d). Using Eq. (10.f) in Eq. (17.b) yields
(17.c)
Augmenting the last equation with Eq. (13) to which, as before, we add , yields
(17.f)
where
(17.g)
Eq. (17.f) is the dynamics model used by the estimator when we apply the implicit approach to gyro calibration. From the explanation of the way this approach works, it is clear that the measurement in the KF sense is just the attitude measurement (and not the gyro measurements), which in our case is simply . From Eq. (15.c) it is obvious that
(18)
Eqs. (17.f) and Eq. (18) constitute, respectively, the dynamics and the measurement equations for the estimator used in the implicit approach to gyro calibration.
5 Compensation
To complete the calibration process we need to perform the compensation stage using the estimated calibration parameters. From Eq. (10.c) we obtain the following estimates of the gyro errors
(19.a)
Since
(19.b)
then a calibrated gyro measurement, , of is computed as follows
(19.c)
6 Implementation Considerations
The dynamics matrix in Eq. (14) is a function of and which are not available. Initially, however, we can evaluate the matrix using rather than but when the attitude converges we can use the estimated quaternion, , which is closer to . Similarly, initially we can use rather than to evaluate this dynamics matrix, the measurement matrix, , of Eq. (15.d), and the dynamics matrix in Eq. (17.f). When in the implicit approach the estimator converges, it is better to switch to the estimated rate denoted by , and when, in the explicit approach the estimate of converges, it is better to use the calibrated rate, , rather than because the calibrated gyro readings are closer to more than is.
7 Test Results
The algorithm was tested using simulated telemetry flight data. Autonomous Star Tracker (AST) and gyro data were simulated. The AST provides the measured quaternion, . The modeled calibration parameter values were:
radians
rad/sec
(The bias values are, respectively, 0.1, -0.2 and 0.3 degrees per second.) Each sensor provided data at a 1 Hz rate. The calibration maneuvers started with a zero inertial rate period of 200 seconds, which was used to estimate the gyro biases. This inertial period was followed by three sequential 0.1 deg/sec maneuvers about the x, y, and z-axes, respectively, lasting 200 seconds each. The run length was 1400 seconds. These calibration maneuvers are shown in Fig.1.
Fig.1: SC rates during the gyro alignment maneuvers.
7.1 Results using the explicit approach
The gyro calibration parameters were estimated well. However, extensive tuning of the filter parameters was required in order to achieve these results. At the end of this simulation the resulting gyro calibration percentage errors, computed as , where was the calibration parameter, were:
Evidently the worst estimation error was below 8%; that is, even in the worst case, no less than 92% of the calibration parameter was estimated.
7.2 Results using the implicit approach
Here too the gyro calibration parameters were estimated well without performing any tuning on the filter parameters. At the end of the simulation the resulting gyro calibration percentage errors, were:
Here the worst estimation error was less than 11%; that is, even in the worst case, no less than 89% of the calibration parameter were estimated.
8. Conclusions
In this paper we compared two filters for in-flight estimation of the calibration parameters of spacecraft gyroscopes. One filter was a realization of the explicit approach and the other was a realization of the implicit approach to gyro calibration.
The filter, which was used in the explicit approach, needed 3 more states than the filter used in the implicit approach, and therefore was more cumbersome. Also that filter required extensive tuning. On the other hand, it was concluded that the performance of both filters was comparable
References:
[1] ASA-GoddardSpaceFlightCenter, Multimission Three-Axis Stabilized Spacecraft (MTASS), 553-FDD-93/032R0UD0, 1933, pp. 3.3.2-1 – 3.3.2-10.
[2] Bar-Itzhack, I.Y., and Harman, R.R., “In-Space Calibration of a Skewed Gyro Quadruplet," AIAA J. of Guidance, Control, and Dynamics, Vol. 25, No. 5, Sept.-Oct. 2002, pp. 852-859.
[3] Bar-Itzhack, I.Y., and Harman, R.R., "Pseudolinear and State-Dependent Riccati Equation Filters for Angular Rate Estimation", AIAA J. of Guidance, Control, and Dynamics,Vol. 22, No. 5, Sept.-Oct. 1999, pp. 723-725. (Engineering Note).
[4] Chatfield, A.B., Fundamentals of High Accuracy Inertial Navigation, Vol. 174
Progress in Astronautics and Aeronautics, AIAA, 1997, pp. 93-106.
[5] Bezooijen, R.W.H., "AST Capabilities," Lockheed Martin Advanced TechnologyCenter, Palo Alto, CA95304-1191. (Slide presentation).
[6] Wertz J.R., (Ed.), Spacecraft Attitude Dynamics and Control, Reidel Publishing Co., Dordrecht, Holland, 1978, p. 512.
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